Dissolve 1 cup salt, 1/2 cup sugar and 3 tablspoons liquid smoke in 4 quarts cold water in a large container. Submerge pork in brine and cover with plastic wrap and refrigerate for 2 hours.
While pork brines, combine mustard and remaining 2 tablespoons liquid smoke in small bowl; set aside. Combine black pepper, smoke paprika, remaining 2 tablspoons sugar and salt, and cayemme pepper in second bowl and set aside.
Preheat oven to 325 degrees.
Remove pork from brine and dry thoroughly with paper towls. Rub mustard mixture over entire surface of the pork. Sprinkle entire surface of each piece with spice mixture. Place pork on wire rack set inside foil lined baking sheet. Place piece of parchment paper over pork, then cover with sheet of aluminum foil, sealing edges to prevent mositure from escaping. Roast pork for 3 hours.
Remove pork from oven, remove and discard parchement paper and foil. Carefully pour off liquid in bottom of baking sheet into fat separator and reserve for sauce. Return pork to oven and cook, uncovered until well browned, tender and temperature registes 200 degrees in instant read thermometer, about 1 1/2 hours. Transfer to serving dish, tent loosely with foil and let rest for 20 minutes.
Using 2 forks, shred pork into bite sized pieces. Toss with 1 cup sauce and season with salt and pepper. Serve with remaining sauce separately.
While pork rests, pour 1/2 cup of defatted cooking liquid from fat separator into medium bowl whisk in remaining ingredients.